The Dupont complex being a \(\mathfrak {g}\) -DG algebra, by the universal property of the Weil algebra
(see [1]}) this connection determines a morphism \(w: W(\mathfrak {g}) \rightarrow \Omega ^*(EG)\) .
In turn, the latter descends to \(K\) -invariants yielding a morphism
\(w_K: W(\mathfrak {g}, K) \rightarrow \Omega ^*(EG/K)\) . Using the contractibility of \(G/K\)  and the
canonical construction of geodesic simplices (cf. [2]} for
metrics of nonpositive curvature, one defines a simplicial
cross-section \(s: BG \rightarrow EG/K\)  (cf. [3]}), which moreover
is equivariant with respect
to the cyclic group action (cf. [3]}). The composition
\( \mu = {\mathcal {I}}_\Delta \circ \, s^*\circ w_K: W(\mathfrak {g}, K) \rightarrow \Omega ^*(NG)\) 
